Develop a lumped parameter model of the following differential equation using Eulers, Huens, and the 4 th order Runga Kutta Method:
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1 Homework 2 Assigned: 2/1/2012 Due: 3/13/2012 Part 1. Comparison of Euler, Huen, and 4 th Order RK methods Develop a lumped parameter model of the following differential equation using Eulers, Huens, and the 4 th order Runga Kutta Method: Validate the model using the following analytical solution: Vary the time step size between 0.05, 0.1,0.2. The total simulation time should be 1.0. Assume that h(t=0) = 0.0 Construct plots of the numerical and analytical solutions as well as the error (exact-analytical). How does the 4 th -order Runga Kutta Method compare with Euler s and Huen s methods? Do these methods appear to converge towards the exact solution as the time step size is reduced? (2)
2 Part 2: Lump Parameter Model of the Isotopic Evolution of Owens Lake Temporal variations in isotopic records of sediment cores from lacustrine and marine environments have been used for decades to reconstruct the climate history of the continents and oceans. Changes in isotopic composition of surface water bodies can be described using lump parameter models of the form presented in lecture. One of the best documented records of the isotopic evolution of a saline lake comes from Friedman et al. (1976) who collected isotopic data from Owens Lake, California. Owens Lake is situated on the Eastern side of the Sierra Nevada near the Mojave Desert. Following an unusually wet winter in which substantial river inflows during the spring snow melt caused an anomalous increase in lake volume. During the subsequent 1.5 years, Friedman et al. sampled the isotopic composition of the lake as well as other pertinent inflows and outflows as the lake returned to its unnatural desiccated state (see the movie Chinatown to find out why). Isotopic Enrichment of Lake water during Evaporation Before we begin our discussion of lake isotope modeling, we need to review the some basic concepts in isotope hydrology. The isotopic composition of a given fluid or solid is usually described in per mill (%o) notation. This is because of the large differences in isotopic abundances of different environmental isotopes. Per mill notation is defined as follows: δ sample =10 3 (1 R sample R stan dard ) (3) R sample and R standard refer to the isotopic ratio of the a given pair of isotopes. For our case, we will be using deuterium ( 2 H): R sample = 2 H sample 1 H sample ;R stan dard = 2 H stan dard 1 H stan dard (4) In our case, the isotopic composition of the standard for dueterium is the isotopic composition for modern ocean water. Standard mean ocean water has an isotopic ratio for deuterium of Isotopes of water respond in a fundamentally different way to evaporation that do dissolved salts. Dissolved cations (e.g. Na 2+ ) and anions (e.g. Cl -1 ) are completely sequestered in the liquid phase upon evaporation. None of the salts ever make it into the vapor phase. For stable isotopes of water such as deuterium ( 2 H) or oxygen ( 18 O) the sequestration is only partial. That is, the heavy isotope of water preferentially remains in the liquid phase while the lighter isotopes of water (i.e. 1 H and 16 O) tend to move to the
3 vapor phase. Under equilibrium conditions, the isotopic enrichment that occurs during evaporation can be described using a fractionation factor: R L = ε LV R V (4) ε LV is the isotopic enrichment factor between the liquid (L) and the vapor (V) phase. In per mill notation, we can express the equilibrium fractionation factor as follows: δ c = ε LV 1+ δ a + δ 100 a (5) δ a is the isotopic composition of atmospheric vapor phase δ c is the isotopic composition of the liquid condensation You will develop a simple model of the isotopic evolution of Owens lake between similar (but not identical) to the one presented by Phillips et al. (1986). Your assignment will be to develop a lumped parameter model of Owens Lake using Euler s method. Owen s Lake is located near Bishop California. It was historically fed by the Owens River (Fig. 2). Historically, Owens Lake was one of the most important stopover sites for migrating waterfowl and shorebirds in the western United States for thousands of years. In spring, thousands of migrating shorebirds move north from wintering areas as far south as Argentina (Patagonia) and Tierra del Fuego. Although water once flowed into the lake from the Owens River, the lake eventually became hypersaline and "possibly the greatest or most intense human-disturbed dust source on earth" (Todd Hinkley, reporting for the U.S. Geological Survey in the mid-1990s) after the Los Angeles Department of Water and Power (LADWP) diverted the lower Owens River to the Los Angeles aqueduct in The governing water balance equation for Owens Lake is given by (Fig. 3): dv l dt = Q p + Q r + Q c Q e Q g (6)
4 V l change in Owens lake volume (m3) Q p volumetric input of precipitation (m 3 ) Q r volumetric input of river input (m 3 ) Q c volumetric input of back condensation (m 3 ) Q g volumetric outflow of groundwater (m 3 ) The flux of the back condensation is related to the evaporative flux as follows: h relative humidity a w is the activity of water Q c = hq v a w (7) Changes in the isotopic evolution of Owens Lake can be expressed as: d( δ l V l ) =δ p Q p +δ r Q r +δ c Q c δ e Q e δ l Q g (8) dt δ l isotopic composition of Owens lake (%o) δ p isotopic composition of precipitation (m 3 ) δ r isotopic composition of river input (m 3 ) δ c isotopic composition of back condensation (m 3 ) A spreadsheet (OwensLakeData.xls) is provided with this assignment with various estimates of influxes, outflows, and deuterium compositions. You will calibrate your model adjusting two poorly constriained parameters: the groundwater flux (Q g ) and the atmospheric composition of deuterium (δ a ). Note the Q g is multiplied by δ l (instead of δ g ) because we will be assuming that this is a perched lake and thus groundwater represents a net loss to the lakes budget. Also, note
5 that the spreadsheet provided calculates many of the pertinent fluxes and isotopic compositions for the inflows and outflows. Note that equation (7) can be recast in a volume based form as follows: dδ l dv l = A Bδ l BV l (9) Note that the fluxes (Q) in the Excel spreadsheet are presented in volumetric form (ie. Q = Δt *Q) thus you don t need to multiple by Δt in equation 6.its already done for you and the product is provided in the spread sheet. Also note that there is an order of magnitude difference in the lake volumes (10 6 m 3 ) and lake fluxes (10 5 m 3 ). Additional details regarding the calculation of the values of the inflows and outflow terms are presented in Phillips et al. (1986) and won t be repeated here. Required: A. Show that equations 8 and 9 are identical. B. Develop a lumped parameter model of the equations (6) and (9). Try to reproduce the observed variations in lake volume and isotopic composition using the fluxes and isotopic composition of the inflows and outflows presented in an Excel spread sheet provided with this lab. Solve for the lake isotopic composition and volume using bi-weekly time steps using Euler method. C. Vary Q g (Column G) and δ a (Column Q) to determine the best fit value of lake volume and lake deuterium composition. D. Can you think of any other parameters or processes that are not well constrained by this model? References Friedman, Smith et al. 1976, Geochimica et Cosmochimica Acta Phillips, F., Person, M., and A. Muller, 1986, A numerical lumped parameter model for simulating the isotopic evolution of closed basin lakes, Journal of Hydrology, v. 85, p
6 Figure 1. Changes in Owens Lake volume and deuterium composition between
7 Figure 2. Images of Owens Lake and salt flats.
8 Figure 3. Schematic diagram illustrating isotopic sources and sinks to Owens Lake.
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